Problem: Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{p^2 + 8p - 9}{p^2 + 9p}$
First factor the expressions in the numerator and denominator. $ \dfrac{p^2 + 8p - 9}{p^2 + 9p} = \dfrac{(p - 1)(p + 9)}{(p)(p + 9)} $ Notice that the term $(p + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p + 9)$ gives: $q = \dfrac{p - 1}{p}$ Since we divided by $(p + 9)$, $p \neq -9$. $q = \dfrac{p - 1}{p}; \space p \neq -9$